Differential Equation question?

This is a particularly simple type of first-order differential equation, known as a "separable" equation.

You have:

y' = dy/dx = y

where y = y(x), i.e., it's a function of x.

We can isolate all occurrences of y on one side of the equation and all occurrences of x on the other side by splitting up the derivative into differentials:

dy/y = dx

Now integrate both sides:

ln(y) - ln(c) = x - d

where ln(c) and d are the constants of integration resulting from the two integrations. Note that because these are just constants, we can combine them into a single constant. This is always the case:

ln(y) - ln(C) = x

where ln(C) = d - ln(c) is just another way of writing the constant.

Now combine the logarithmic terms using standard rules for manipulating logs.

ln(y/C) = x

Exponentiate both sides:

y/C = exp(x)

y(x) = C*exp(x)